Answer :
To determine which expression is NOT equal to [tex]\(\frac{3 \sqrt{s^5}}{6 \sqrt{s^2}}\)[/tex], we will simplify each option and compare them with the original expression.
First, let's simplify the original expression:
[tex]\[ \frac{3 \sqrt{s^5}}{6 \sqrt{s^2}} = \frac{3 \cdot s^{5/2}}{6 \cdot s^1} = \frac{3 \cdot s^{5/2}}{6 \cdot s^1} = \frac{3 \cdot s^{5/2}}{6 \cdot s} = \frac{3 \cdot s^{5/2}}{6 \cdot s^{2/2}} = \frac{3 \cdot s^{5/2}}{6 \cdot s^{1/2}} \][/tex]
Now, simplify further:
[tex]\[ \frac{3 \cdot s^{5/2}}{6 \cdot s^{1/2}} = \frac{3}{6} \cdot \frac{s^{5/2}}{s^{1/2}} = \frac{1}{2} \cdot s^{(5/2 - 1/2)} = \frac{1}{2} \cdot s^2 = \frac{s^2}{2} \][/tex]
So, [tex]\(\frac{3 \sqrt{s^5}}{6 \sqrt{s^2}} = \frac{s^2}{2}\)[/tex].
Next, let's simplify each option and compare them to the original simplified form [tex]\(\frac{s^2}{2}\)[/tex].
Option A: [tex]\(\frac{3 s^2 \sqrt{s}}{6 \sqrt{s^2}}\)[/tex]
[tex]\[ \frac{3 s^2 \sqrt{s}}{6 \sqrt{s^2}} = \frac{3 s^2 \cdot s^{1/2}}{6 \cdot s^1} = \frac{3 s^{5/2}}{6 s} = \frac{3 s^{5/2}}{6 s^{2/2}} = \frac{3 s^{5/2}}{6 s^{1/2}} = \frac{3}{6} \cdot \frac{s^{5/2}}{s^{1/2}} = \frac{1}{2} \cdot s^2 = \frac{s^2}{2} \][/tex]
So, Option A is equal to [tex]\(\frac{s^2}{2}\)[/tex].
Option B: [tex]\(\frac{s^2}{2 \sqrt{s^2}}\)[/tex]
[tex]\[ \frac{s^2}{2 \sqrt{s^2}} = \frac{s^2}{2 \cdot s} = \frac{s^2}{2 s} = \frac{s}{2} \][/tex]
So, Option B is [tex]\(\frac{s}{2}\)[/tex], which is NOT equal to [tex]\(\frac{s^2}{2}\)[/tex].
Option C: [tex]\(\frac{s^2}{2 \sqrt{s}}\)[/tex]
[tex]\[ \frac{s^2}{2 \sqrt{s}} = \frac{s^2}{2 \cdot s^{1/2}} = \frac{s^{2 - 1/2}}{2} = \frac{s^{3/2}}{2} \][/tex]
So, Option C is [tex]\(\frac{s^{3/2}}{2}\)[/tex], which is also NOT equal to [tex]\(\frac{s^2}{2}\)[/tex].
Option D: [tex]\(\frac{s \sqrt{s}}{2}\)[/tex]
[tex]\[ \frac{s \sqrt{s}}{2} = \frac{s \cdot s^{1/2}}{2} = \frac{s^{3/2}}{2} \][/tex]
So, Option D is [tex]\(\frac{s^{3/2}}{2}\)[/tex], which is also NOT equal to [tex]\(\frac{s^2}{2}\)[/tex].
Option E: [tex]\(\frac{\sqrt{s^3}}{2}\)[/tex]
[tex]\[ \frac{\sqrt{s^3}}{2} = \frac{s^{3/2}}{2} \][/tex]
So, Option E is [tex]\(\frac{s^{3/2}}{2}\)[/tex], which is also NOT equal to [tex]\(\frac{s^2}{2}\)[/tex].
Hence, the option that is NOT equal to [tex]\(\frac{3 \sqrt{s^5}}{6 \sqrt{s^2}}\)[/tex] (or [tex]\(\frac{s^2}{2}\)[/tex]) is Option (B): [tex]\(\frac{s^2}{2 \sqrt{s^2}}\)[/tex].
First, let's simplify the original expression:
[tex]\[ \frac{3 \sqrt{s^5}}{6 \sqrt{s^2}} = \frac{3 \cdot s^{5/2}}{6 \cdot s^1} = \frac{3 \cdot s^{5/2}}{6 \cdot s^1} = \frac{3 \cdot s^{5/2}}{6 \cdot s} = \frac{3 \cdot s^{5/2}}{6 \cdot s^{2/2}} = \frac{3 \cdot s^{5/2}}{6 \cdot s^{1/2}} \][/tex]
Now, simplify further:
[tex]\[ \frac{3 \cdot s^{5/2}}{6 \cdot s^{1/2}} = \frac{3}{6} \cdot \frac{s^{5/2}}{s^{1/2}} = \frac{1}{2} \cdot s^{(5/2 - 1/2)} = \frac{1}{2} \cdot s^2 = \frac{s^2}{2} \][/tex]
So, [tex]\(\frac{3 \sqrt{s^5}}{6 \sqrt{s^2}} = \frac{s^2}{2}\)[/tex].
Next, let's simplify each option and compare them to the original simplified form [tex]\(\frac{s^2}{2}\)[/tex].
Option A: [tex]\(\frac{3 s^2 \sqrt{s}}{6 \sqrt{s^2}}\)[/tex]
[tex]\[ \frac{3 s^2 \sqrt{s}}{6 \sqrt{s^2}} = \frac{3 s^2 \cdot s^{1/2}}{6 \cdot s^1} = \frac{3 s^{5/2}}{6 s} = \frac{3 s^{5/2}}{6 s^{2/2}} = \frac{3 s^{5/2}}{6 s^{1/2}} = \frac{3}{6} \cdot \frac{s^{5/2}}{s^{1/2}} = \frac{1}{2} \cdot s^2 = \frac{s^2}{2} \][/tex]
So, Option A is equal to [tex]\(\frac{s^2}{2}\)[/tex].
Option B: [tex]\(\frac{s^2}{2 \sqrt{s^2}}\)[/tex]
[tex]\[ \frac{s^2}{2 \sqrt{s^2}} = \frac{s^2}{2 \cdot s} = \frac{s^2}{2 s} = \frac{s}{2} \][/tex]
So, Option B is [tex]\(\frac{s}{2}\)[/tex], which is NOT equal to [tex]\(\frac{s^2}{2}\)[/tex].
Option C: [tex]\(\frac{s^2}{2 \sqrt{s}}\)[/tex]
[tex]\[ \frac{s^2}{2 \sqrt{s}} = \frac{s^2}{2 \cdot s^{1/2}} = \frac{s^{2 - 1/2}}{2} = \frac{s^{3/2}}{2} \][/tex]
So, Option C is [tex]\(\frac{s^{3/2}}{2}\)[/tex], which is also NOT equal to [tex]\(\frac{s^2}{2}\)[/tex].
Option D: [tex]\(\frac{s \sqrt{s}}{2}\)[/tex]
[tex]\[ \frac{s \sqrt{s}}{2} = \frac{s \cdot s^{1/2}}{2} = \frac{s^{3/2}}{2} \][/tex]
So, Option D is [tex]\(\frac{s^{3/2}}{2}\)[/tex], which is also NOT equal to [tex]\(\frac{s^2}{2}\)[/tex].
Option E: [tex]\(\frac{\sqrt{s^3}}{2}\)[/tex]
[tex]\[ \frac{\sqrt{s^3}}{2} = \frac{s^{3/2}}{2} \][/tex]
So, Option E is [tex]\(\frac{s^{3/2}}{2}\)[/tex], which is also NOT equal to [tex]\(\frac{s^2}{2}\)[/tex].
Hence, the option that is NOT equal to [tex]\(\frac{3 \sqrt{s^5}}{6 \sqrt{s^2}}\)[/tex] (or [tex]\(\frac{s^2}{2}\)[/tex]) is Option (B): [tex]\(\frac{s^2}{2 \sqrt{s^2}}\)[/tex].
